(a)
\(\mu ^A\)  is the unique measure of minimum potential in the class \(\Gamma _{A,\mu }^+\) , that is, \(\mu ^A\in \Gamma _{A,\mu }^+\)  and
\(\kappa \mu ^A=\min _{\nu \in \Gamma _{A,\mu }^+}\,\kappa \nu \text{ \ on $X$}.\) 
(b)
\(\mu ^A\)  is the unique orthogonal projection of \(\mu \)  in the pre-Hilbert space \(\mathcal {E}\)  onto the (convex, strongly complete) cone \(\mathcal {E}^{\prime }_A\) , that is, \(\mu ^A\in \mathcal {E}^{\prime }_A\)  andConcerning the concept of orthogonal projection, see e.g. [1]}. (Observe that, being a strongly closed subset of the strongly complete cone \(\mathcal {E}^+\) , \(\mathcal {E}^{\prime }_A\)  is likewise strongly complete.)
\(\Vert \mu -\mu ^A\Vert =\min _{\nu \in \mathcal {E}_A^{\prime }}\,\Vert \mu -\nu \Vert .\) 
(c)
\(\mu ^A\)  is the unique measure in \(\mathcal {E}^{\prime }_A\)  having the property
\(\kappa \mu ^A=\kappa \mu \text{ \ n.e.\ on\ }A.\) 
(d)
\(\mu ^A\)  is the unique measure in \(\mathcal {E}^+\)  satisfying any of the three limit relations
\(&\mu ^K\rightarrow \mu ^A\text{ \ strongly in $\mathcal {E}^+$ as $K\uparrow A$},\\&\mu ^K\rightarrow \mu ^A\text{ \ vaguely in $\mathcal {E}^+$ as $K\uparrow A$},\\&\kappa \mu ^K\uparrow \kappa \mu ^A\text{ \ pointwise on $X$ as $K\uparrow A$},\) 

where \(\mu ^K\)  denotes the only measure in \(\mathcal {E}^+_K\)  with \(\kappa \mu ^K=\kappa \mu \)  n.e. on \(K\) .


